3.274 \(\int \frac{e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{2 \sqrt{c-a c x}}{a c^2 \sqrt{1-a^2 x^2}} \]

[Out]

(-2*Sqrt[c - a*c*x])/(a*c^2*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.0504776, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6127, 649} \[ -\frac{2 \sqrt{c-a c x}}{a c^2 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^(3/2)),x]

[Out]

(-2*Sqrt[c - a*c*x])/(a*c^2*Sqrt[1 - a^2*x^2])

Rule 6127

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^n, Int[(c + d*x)^(p - n)*(1 -
a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]

Rule 649

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rubi steps

\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=\frac{\int \frac{(c-a c x)^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac{2 \sqrt{c-a c x}}{a c^2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0235171, size = 35, normalized size = 1.06 \[ -\frac{2 (1-a x)^{3/2}}{a \sqrt{a x+1} (c-a c x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^(3/2)),x]

[Out]

(-2*(1 - a*x)^(3/2))/(a*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))

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Maple [A]  time = 0.031, size = 34, normalized size = 1. \begin{align*} -2\,{\frac{ \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3/2}}{ \left ( -acx+c \right ) ^{3/2} \left ( ax+1 \right ) ^{2}a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(3/2),x)

[Out]

-2*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(3/2)/(a*x+1)^2/a

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Maxima [A]  time = 0.984977, size = 38, normalized size = 1.15 \begin{align*} -\frac{2 \, \sqrt{a x + 1} \sqrt{c}}{a^{2} c^{2} x + a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(3/2),x, algorithm="maxima")

[Out]

-2*sqrt(a*x + 1)*sqrt(c)/(a^2*c^2*x + a*c^2)

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Fricas [A]  time = 1.57041, size = 82, normalized size = 2.48 \begin{align*} \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a^{3} c^{2} x^{2} - a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(3/2),x, algorithm="fricas")

[Out]

2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^3*c^2*x^2 - a*c^2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(3/2),x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.19356, size = 41, normalized size = 1.24 \begin{align*} \frac{{\left (\frac{\sqrt{2}}{a \sqrt{c}} - \frac{2}{\sqrt{a c x + c} a}\right )}{\left | c \right |}}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^(3/2),x, algorithm="giac")

[Out]

(sqrt(2)/(a*sqrt(c)) - 2/(sqrt(a*c*x + c)*a))*abs(c)/c^2